Random Initialization Solves Shapley's Fictitious Play Counterexample

TL;DR

This paper demonstrates that random initializations in fictitious play can overcome Shapley's classic counterexample, leading to convergence to Nash equilibrium approximately one-third of the time.

Contribution

It introduces the idea that random initial strategy profiles can resolve convergence issues in Shapley's fictitious play counterexample.

Findings

Random initializations lead to convergence in about 1/3 of cases.

Fictitious play can be made effective with randomized starting points.

The approach offers a practical solution to Shapley's counterexample.

Abstract

In 1964 Shapley devised a family of games for which fictitious play fails to converge to Nash equilibrium. The games are two-player non-zero-sum with 3 pure strategies per player. Shapley assumed that each player played a specific pure strategy in the first round. We show that if we use random (mixed) strategy profile initializations we are able to converge to Nash equilibrium approximately 1/3 of the time for a representative game in this class.